I bet the laptop connects direct to the printer without using the router.
I'm pretty sure this is a direct wifi connection printer in that you don't need a router to connect to wireless devices. On that basis your desktop is never going to connect without it's own wifi card.

Check the Port the Printer is using, in Printer Properties, and make sure it is using the correct IP address. You can compare the ports on the laptop w/the desktop to make sure they are setup the same, i.e. they should both be using the same IP address or IP port the printer software establishes. PrinterIPPort.JPG

Your computer does not need a wireless card. The problem is with the communications to the printer. It could be something as simple as the IP address or port. It could also be something as complicated as your PC using an IP packet size incompatible with the printer's connection.

The first thing I would try would be to do the setup from the problem PC. Hopefully the manufacturer checks things like packet size to ensure it works.

If that doesn't fix the problem,then Email the manufacturer with the message and have them walk you thru the setup. They probably ran into this problem before so it would be on their cheat sheet used to fix this timeout error.